Chapter 4.7, Problem 25ES

The mechanic’s rule for approximating square roots states that $\sqrt{a}\approx x_{n+1},$ where $x_{n+1}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right),n=1,2,3,\dots$ and $x_1$ is any positive approximation to $\sqrt{a}$ .

(a) Apply Newton’s Method to $f\left(x\right)=x^2-a$ to derive the mechanic’s rule.

(b) Use the mechanic’s rule to approximate $\sqrt{10}$ .